Sur l’existence d’une solution ramifiée pour des équations de Fuchs à caractéristique simple

Source abstract

The aim of this paper is to construct a holomorphic solution, ramified around a simple characteristic hypersurface, for some linear Fuchsian equation of order m ≥ 1 m\geq 1 . We consider an operator L L , holomorphic in a neighborhood of the origin in C t × C x n {\mathbb {C}}_t\times {\mathbb {C}}_x^n , of the form L = t A + B L=tA+B where A A and B B are linear partial differential operators of order m m and m − 1 m-1 , and where A A has a simple characteristic hypersurface transverse to S : t = 0 S:t=0 . Under an assumption linking the principal symbols of A A and B B , the question is reduced to the study of an integro-differential Fuchsian equation with an additional variable z z that describes the universal covering of a pointed disk. It is an equation where terms like t l D t h D x α ( t D t + 1 ) − 1 D z − q , l , h , q ∈ N , α ∈ N n t^lD_t^hD_x^\alpha (tD_t+1)^{-1}D_z^{-q}, l,h,q\in \mathbb {N}, \alpha \in \mathbb {N}^n with l ≤ 1 l\leq 1 and h + | α | ≤ l + q h+|\alpha |\leq l+q appear. The problem is solved by the fixed-point theorem with appropriate estimations in a Banach space.